The Stacks project

Proof. It is important in this proof to argue for all quasi-compact opens $U \subset X$ at the same time. The result is true for $q = 0$ and any quasi-compact open $U \subset X$ by Sheaves, Lemma 6.29.1 (combined with Topology, Lemma 5.27.1). Assume that we have proved the result for all $q \leq q_0$ and let us prove the result for $q = q_0 + 1$.

By our conventions on directed systems the index set $I$ is directed, and any system of $\mathcal{O}_ X$-modules $(\mathcal{F}_ i, \varphi _{ii'})$ over $I$ is directed. By Injectives, Lemma 19.5.1 the category of $\mathcal{O}_ X$-modules has functorial injective embeddings. Thus for any system $(\mathcal{F}_ i, \varphi _{ii'})$ there exists a system $(\mathcal{I}_ i, \varphi _{ii'})$ with each $\mathcal{I}_ i$ an injective $\mathcal{O}_ X$-module and a morphism of systems given by injective $\mathcal{O}_ X$-module maps $\mathcal{F}_ i \to \mathcal{I}_ i$. Denote $\mathcal{Q}_ i$ the cokernel so that we have short exact sequences

We claim that the sequence

is also a short exact sequence of $\mathcal{O}_ X$-modules. We may check this on stalks. By Sheaves, Sections 6.28 and 6.29 taking stalks commutes with colimits. Since a directed colimit of short exact sequences of abelian groups is short exact (see Algebra, Lemma 10.8.8) we deduce the result. We claim that $H^ q(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = 0$ for all quasi-compact open $U \subset X$ and all $q \geq 1$. Accepting this claim for the moment consider the diagram

The zero at the lower right corner comes from the claim and the zero at the upper right corner comes from the fact that the sheaves $\mathcal{I}_ i$ are injective. The top row is exact by an application of Algebra, Lemma 10.8.8. Hence by the snake lemma we deduce the result for $q = q_0 + 1$.

It remains to show that the claim is true. We will use Lemma 20.11.9. Let $\mathcal{B}$ be the collection of all quasi-compact open subsets of $X$. This is a basis for the topology on $X$ by assumption. Let $\text{Cov}$ be the collection of finite open coverings $\mathcal{U} : U = \bigcup _{j = 1, \ldots , m} U_ j$ with each of $U$, $U_ j$ quasi-compact open in $X$. By the result for $q = 0$ we see that for $\mathcal{U} \in \text{Cov}$ we have

because all the multiple intersections $U_{j_0 \ldots j_ p}$ are quasi-compact. By Lemma 20.11.1 each of the complexes in the colimit of Čech complexes is acyclic in degree $\geq 1$. Hence by Algebra, Lemma 10.8.8 we see that also the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i)$ is acyclic in degrees $\geq 1$. In other words we see that $\check{H}^ p(\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = 0$ for all $p \geq 1$. Thus the assumptions of Lemma 20.11.9 are satisfied and the claim follows. $\square$

Proof. Recall that $R^ pf_*\mathcal{F}$ is the sheafification of the presheaf $\mathcal{G}$ sending $V$ to $H^ p(f^{-1}(V), \mathcal{F})$, see Lemma 20.7.3. Similarly, $R^ pf_*\mathcal{F}_ i$ is the sheafification of the presheaf $\mathcal{G}_ i$ sending $V$ to $H^ p(f^{-1}(V), \mathcal{F}_ i)$. Recall that sheafification is the left adjoint to the inclusion from sheaves to presheaves, see Sheaves, Section 6.17. Hence sheafification commutes with colimits, see Categories, Lemma 4.24.5. Hence it suffices to show that the map of presheaves (with colimit in the category of presheaves)

induces an isomorphism on sheafifications. For this it suffices to show that the presheaves $\mathcal{G}$ and $\mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ agree on a basis for the topology of $Y$. Namely, in this case the stalks of their sheafifications, which can be computed directly from the presheaf values on elements of the basis, agree. The required agreement is exactly the assumption of the lemma. $\square$

Proof. The case $p = 0$ is Sheaves, Lemma 6.29.4.

In this paragraph we show that we can find a map of systems $(\gamma _ i) : (\mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ with $\mathcal{G}_ i$ an injective abelian sheaf and $\gamma _ i$ injective. For each $i$ we pick an injection $\mathcal{F}_ i \to \mathcal{I}_ i$ where $\mathcal{I}_ i$ is an injective abelian sheaf on $X_ i$. Then we can consider the family of maps

where the component maps are the maps adjoint to the maps $f_ b^{-1}\mathcal{F}_ i \to \mathcal{F}_ k \to \mathcal{I}_ k$. For $a : j \to i$ in $\mathcal{I}$ there is a canonical map

whose components are the canonical maps $f_ b^{-1}f_{a \circ b, *}\mathcal{I}_ k \to f_{b, *}\mathcal{I}_ k$ for $b : k \to j$. Thus we find an injection $\{ \gamma _ i\} : \{ \mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ of systems of abelian sheaves. Note that $\mathcal{G}_ i$ is an injective sheaf of abelian groups on $X_ i$, see Lemma 20.11.11 and Homology, Lemma 12.27.3. This finishes the construction.

Arguing exactly as in the proof of Lemma 20.19.1 we see that it suffices to prove that $H^ p(X, \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{G}_ i) = 0$ for $p > 0$.

Set $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{G}_ i$. To show vanishing of cohomology of $\mathcal{G}$ on every quasi-compact open of $X$, it suffices to show that the Čech cohomology of $\mathcal{G}$ for any open covering $\mathcal{U}$ of a quasi-compact open of $X$ by finitely many quasi-compact opens is zero, see Lemma 20.11.9. Such a covering is the inverse by $p_ i$ of such a covering $\mathcal{U}_ i$ on the space $X_ i$ for some $i$ by Topology, Lemma 5.24.6. We have

by the case $p = 0$. The right hand side is a filtered colimit of complexes each of which is acyclic in positive degrees by Lemma 20.11.1. Thus we conclude by Algebra, Lemma 10.8.8. $\square$